Say you have a number xyz and you choose to split it to digits, take a power of each digit to three and summarize them. Some numbers gives same result than the original number, for example:
153 = 1^3 + 5^3 + 3^3 = 153
4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7 = 4210818
Then some numbers gives other number, but when you recursively create a new number with the same algorithm from the result, you tend to find out certain list of numbers that are reductive end numbers. Like:
1080 => 1^3 + 0^3 + 8^3 + 0^3 = 513 => 5^3 + 1^3 + 3^3 = 153 = 1^3 + 5^3 + 3^3
Where 153 is the reductive end number.
For the power of 3 I have found 15 reductive numbers (1, 55, 133, 136, 153, 160, 217, 244, 250, 352, 370, 371, 407, 919, 1459). Most of them reduces to 153 & 371. I cannot find any new reductive pow numbers after 479 (tried with a limit of 50000). So the smallest reduce able number (SRN) to provide all reductive pow numbers is 479. Maximum recursive iteration needed to find out reductive pow numbers is 14 with a given limit.
My questions are: Is there any mathematical way to prove that there are no more reductive pow numbers after n? What kind of mathematical problem domain this is, something to do with probability or predictability or what?
I have created a table of pows from 2-9 that gives a summary of the results of the algorithm applied to certain limit of numbers.
EXP CORN MPRN HRV MIOR SRN LIMIT
2 9 89 145 19 145 50000
3 15 153 1459 14 478 50000
4 13 13139 13139 58 11339 50000
5 95 9045 213040 75 58999 100000
6 39 383890 1458364 100 47778 100000
7 129 5905147 14628971 159 57889 100000
8 84 9514916 153362052 259 78999 100000
9 129 389778106 1430717631 186 66699 100000
EXP = Exponent / Power CORN = Count of reduced numbers found between 1 and n (limit) MPRN = Most prominent reduced number, number that occurs most often as a result of reduction HRV = Highest reduced value MIOR = Maximum iterations occured on reductions SRN = Smallest reduce able number to find out all reduced values between 1 and n (limit) LIMIT = Highest number to reduce
Finally I'd like to provide a picture to show the pattern of reduced pow 3 numbers:
